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Full text of "Analytical Mechanics"

114 ANALYTICAL MECHANICS 103. Equations of Motion. — The force equation and the equations (1), (2), and (3), which connect v, s, and t are called equations of motion. The force equation will be called the differential equation of motion, while those which are obtained by integrating the force equation will be called the integral equations of motion. 104. Special Cases: A. Motion when the Force is Zero. — When the force vanishes the acceleration is zero. Therefore equations (1) to (3) become i) = VQ = const., s = vQL Therefore the particle moves in a straight path with unchanging velocity. 105. B. Falling Bodies. — The force experienced by a falling body is its weight mg. Therefore the acceleration of the motion is g, the gravitational acceleration due to the attraction of the earth. So long as the distance through which the body falls is very small compared with the radius of the earth, g may be considered to remain constant. Therefore the motion of falling bodies may be treated as a special case of rectilinear motion under a constant force. Hence the equations of motion of a falling body are obtained by replacing / by g in equations (1) to (3). Making this substitution we get v = vQ+gt, v* = vQ* + 2gs. When a body falls from rest the initial velocity is zero, Therefore we must put VQ = 0 in the last three equations before using them for bodies falling from rest. When a body is projected vertically upward the acceleration is in the opposite direction from the velocity; in other words, it is negative. Therefore in the last three equations